Department of Computer Engineering

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Department of Computer Engineering
ARTI Artificial Intelligence Laboratory
Member Murat ERENTÜRK
Performance Analysis of Meta-Heuristic Approaches
for Traveling Salesperson Problem
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Department of Computer Engineering
Problem A salesman must visit every city in his
territory exactly once and return to his starting
city. Given the cost of travel between all
cities, how should he plan his itinerary to
minimize the total cost of his tour?
Performance Analysis of Meta-Heuristic Approaches
for Traveling Salesperson Problem
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• Search space is a set of permutations of n cities
• Traditional methods fail (tree search) size is
  n!
• No polinomial time algorithm exists
• Problem is considered NP-complete
• Many heuristics developed
• Has many applications

Performance Analysis of Meta-Heuristic Approaches
for Traveling Salesperson Problem
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PROBLEM REPRESNETATION
Performance Analysis of Meta-Heuristic Approaches
for Traveling Salesperson Problem
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All possible configurations have to be considered
to find the optimal solution. For our example the
possible solutions are a, b, c, d a, b,
d, c a, c, d, b a, d, b, c a, d, c, b
a, c, b, d
As we can see for N cities there are (N -1)!
possible configurations
Performance Analysis of Meta-Heuristic Approaches
for Traveling Salesperson Problem
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• When N is small (like in our example) its very
  easy to find the optimal solution. However, when
  N becomes large the need for a computer becomes
  useful
• If we assume a computer can evaluate 1010
  configurations in a second, how long would it
  take to evaluate all the possible solutions of N?

Performance Analysis of Meta-Heuristic Approaches
for Traveling Salesperson Problem
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Performance Analysis of Meta-Heuristic Approaches
for Traveling Salesperson Problem
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• Nearly 2 million years just for 25 cities. This
  exponential growth in computation time makes
  enumeration of all possible solutions a
  non-starter.
• So we have to look for different kind of
  approaches

Performance Analysis of Meta-Heuristic Approaches
for Traveling Salesperson Problem
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Heuristics

• A heuristic is a method used to guess the cost of
  getting from a non-goal state to a goal state.

• Heuristics are good when
• you have to make a moment decision OR
• you have limited information and cannot obtain
  more OR
• the decision is not that important
• Heuristics are bad when
• you have plenty of time and information to make
  an important decision
• you need to be right 100 of the time

Performance Analysis of Meta-Heuristic Approaches
for Traveling Salesperson Problem
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2-OPT
A 2-opt move consists of eliminating two edges
and reconnecting the two resulting paths in a
different way to obtain a new tour. This is a
method by which we can improve the old tour by
the way of exchanging better neighbor than the
old neighbor. The main idea is to take all
nonadjacent vertices of the tour on the graph and
take the distance between them and their adjacent
and compare with the old distance
Performance Analysis of Meta-Heuristic Approaches
for Traveling Salesperson Problem
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Let us suppose that these edges are (a, b) and
(c, d). Then we remove edges (a, b) and (c, d)
and swap b with c in other words edges become
(a, c) and (b, d). This operation is called a
2-interchange.
n(n-1)/2
. If the resulting solution is smaller than the
initial solution, it is stored as a candidate for
future consideration. Thus, whenever a better
solution is found, the algorithm discards the
previous best solution. If not, it is discarded
and the algorithm continues the iteration.
n(n-1)/2 replacements made
Performance Analysis of Meta-Heuristic Approaches
for Traveling Salesperson Problem
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Let us suppose that these edges are (a, b) and
(c, d). Then we remove edges (a, b) and (c, d)
and swap b with c in other words edges become
(a, c) and (b, d). This operation is called a
2-interchange.
n(n-1)/2
. If the resulting solution is smaller than the
initial solution, it is stored as a candidate for
future consideration. Thus, whenever a better
solution is found, the algorithm discards the
previous best solution. If not, it is discarded
and the algorithm continues the iteration.
n(n-1)/2 replacements made
Performance Analysis of Meta-Heuristic Approaches
for Traveling Salesperson Problem
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• 2-OPT ANIMATION

Performance Analysis of Meta-Heuristic Approaches
for Traveling Salesperson Problem
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Hill Climbing (Local Search Method)
A neighbourhood search or so called local search
method that starts from some initial solution and
moves to a better neighbouring solution until it
arrives at a local optimum
Performance Analysis of Meta-Heuristic Approaches
for Traveling Salesperson Problem
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Hill-climbing Algorithm

• Pick a random point in the search space
• Consider all the neighbour of the current state
• Choose the neighbour with the best quality and
  move to that state
• Repeat 2 thru 4 until all the neighbouring states
  are of lower quality
• Return the current state as the solution state

Performance Analysis of Meta-Heuristic Approaches
for Traveling Salesperson Problem
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Hill climbing algorithm has three well-known
drawbacks as given below

• Local Optimum All neighboring states are worse
  or the same. The algorithm will halt even though
  the solution may be far from satisfactory.
• Plateau All neighboring states are the same as
  the current state. In other words the evaluation
  function is essentially flat. The search will
  conduct a random walk
• Ridge The search may oscillate from side to
  side, making little progress. In each case, the
  algorithm reaches a point at which no progress is
  being made. If this happens, an obvious thing to
  do is start again from a different starting
  point.
• As a result may not find the minimum cost
  solution and may get stuck in local minima

Performance Analysis of Meta-Heuristic Approaches
for Traveling Salesperson Problem
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• Meta-Heuristics

These algorithms guide an underlying
heuristic/local search to escape from being
trapped in a local optima and to explore better
areas of the solution space
Performance Analysis of Meta-Heuristic Approaches
for Traveling Salesperson Problem
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Meta-Heuristic Approaches

• The most commonly used meta heuristic approaches
  for solving TSP are
• Genetic Algorithms (GA)
• Simulated Annealing (SA)
• Memetic Algorithms GA Hill Climbing
• Memetic Annealing SA Hill Climbing
• Tabu Search

Performance Analysis of Meta-Heuristic Approaches
for Traveling Salesperson Problem
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Simulated Annealing (SA)

• SA is an analogy with thermodynamics,
  specifically with the way that metals cool and
  anneal.
• At high temperatures, the molecules move freely
  with respect to one another. If the liquid is
  cooled slowly, thermal mobility is lost. The
  amazing fact is that, for slowly cooled systems,
  nature is able to find this minimum energy state.
  
• According to the idea that a system in thermal
  equilibrium at temperature T has its energy
  probabilistically distributed among all different
  energy states E (Boltzmann probability
  distribution).

Performance Analysis of Meta-Heuristic Approaches
for Traveling Salesperson Problem
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When stuck on a local minimum just like in
hill-climbing, we could allow the search to take
some uphill steps to escape the local minimum
Performance Analysis of Meta-Heuristic Approaches
for Traveling Salesperson Problem
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• Generate the initial configuration
• Tcurrent temperature
• Do i1,k
• Apply neighborhood operator, obtaining a new
  configuration
• Calculate the change in energy, dE E1-E2
• If (dE?0) then
• its a downhill move to lower energy so accept
  and update configuration
• Else
• its an uphill move so generate random number
  P0,1
• if (PltPr(dE) then // compare with
  Pr(dE)exp(-dE/kT)
• accept move and update configuration
• Else
• reject move keep original configuration

Performance Analysis of Meta-Heuristic Approaches
for Traveling Salesperson Problem
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• Cooling Schedule Strategy
• Linear
• temp temp - a
• Geometric
• temp temp a

a should be between 0.8 and 0.99, with better
results being found in the higher end of the
range. Of course, the higher the value of a, the
longer it will take to decrement the temperature
to the stopping criterion
Performance Analysis of Meta-Heuristic Approaches
for Traveling Salesperson Problem
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Neighbourhood operators used
Mutation operators used as implemented in genetic
algorithms for Neighbourhood operator
Performance Analysis of Meta-Heuristic Approaches
for Traveling Salesperson Problem
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Genetic Algorithms (GAs)

• GAs (John Holland) simulate natural evolution
  (Darwinian Evolution) at the genetic level using
  the idea of survival of the fittest
• An individual (chromosome) represents a candidate
  solution for the problem at hand. A collection of
  individuals currently "alive, called population
  is evolved from one generation to another
  depending on the fitness of individuals,
  indicating how fit an individual is, in other
  words, how close it is to the solution.
• Hope Last generation will contain the final
  solution

Performance Analysis of Meta-Heuristic Approaches
for Traveling Salesperson Problem
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Genetic Algorithm

• Generate the initial generation with the
  population P(0), let i indicate the generation
• Repeat until the population converges or a
  termination criteria is satisfied
• Evaluate the fitness of each individual in P(i)
• Select parents from P(i) based on their fitness
  in P(i)
• Apply genetic operators (crossover, mutation) to
  the parents and produce offspring
• Obtain the next generation P(i 1) from
  offspring and the individuals in the generation
  P(i)

Performance Analysis of Meta-Heuristic Approaches
for Traveling Salesperson Problem
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Basic Components of GAs

• Representation
• Initial population generation - Initialization
• Fitness Function
• Selection for recombination
• Crossover
• Mutation
• Replacement Strategy
• Termination Criteria

Performance Analysis of Meta-Heuristic Approaches
for Traveling Salesperson Problem
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Fitness value
evaluation
Population
001100101010
Calibration model
Variable set
RMS
Selection Crossover Mutation
Generate new population set
Performance Analysis of Meta-Heuristic Approaches
for Traveling Salesperson Problem
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Structure of GA
Step 0 Initialization

• Procedure GA
• begin
• t 0
• initialize P(t)
• evaluate P(t)
• while (not termination-condition) do
• begin
• t t 1
• select P(t) from P(t-1)
• alter P(t)
• evaluate P(t)
• end
• end

Step 1 Selection
Step 2 Crossover
Step 3 Mutation
Step 4 Evaluation
Step 5 Termination Test
Step 6 End
Performance Analysis of Meta-Heuristic Approaches
for Traveling Salesperson Problem
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Path Representation
Performance Analysis of Meta-Heuristic Approaches
for Traveling Salesperson Problem
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Initialization
Performance Analysis of Meta-Heuristic Approaches
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Fitness Function
f(indiv)S d i (i1) dn1 1?i lt n
Performance Analysis of Meta-Heuristic Approaches
for Traveling Salesperson Problem
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• GA ANIMATION

Performance Analysis of Meta-Heuristic Approaches
for Traveling Salesperson Problem
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Selection - Crossover

• Selection in GAs aims at giving higher chance for
  fitter individuals to be selected as parents to
  produce even better offspring
• Rank-based Selection
• Tournament Selection
• Crossover exchanges parts from parent individuals
  producing new offspring
• PMX
• k-PTX
• Uniform Crossover

Performance Analysis of Meta-Heuristic Approaches
for Traveling Salesperson Problem
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PMX XOVER

• builds offspring by
• choose a subsequence of a tour from one parent
• choose two random cut points to serve as swapping
  boundaries
• swap segments between cut points
• preserve the order and position of as many cities
  as possible from other parent

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Example

• p1 (1 2 3 4 5 6 7 8 9)
• p2 (4 5 2 1 8 7 6 9 3)
• step 1 swap segments
• o1 (x x x 1 8 7 6 x x)
• o2 (x x x 4 5 6 7 x x)
• this also defines mappings
• 1?4, 8?5, 7?6, 6?7
• step 2 fill in cities from other parents if no
  conflict
• o1 (x 2 3 1 8 7 6 x 9)
• o2 (x x 2 4 5 6 7 9 3)
• step 3 use mappings for conflicted positions
• o1 (4 2 3 1 8 7 6 5 9)
• o2 (1 8 2 4 5 6 7 9 3)

Performance Analysis of Meta-Heuristic Approaches
for Traveling Salesperson Problem
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OX XOVER

• builds offspring by
• choose a subsequence of a tour from one parent
• preserve relative order of cities from other
  parent

Performance Analysis of Meta-Heuristic Approaches
for Traveling Salesperson Problem
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Example

• p1 (1 2 3 4 5 6 7 8 9)
• p2 (4 5 2 1 8 7 6 9 3)
• step 1 copy segments into offspring
• o1 (x x x 4 5 6 7 x x)
• o2 (x x x 1 8 7 6 x x)
• step 2 starting from 2nd cut point of one
  parent, cities from other parent copied in same
  order, omitting symbols already present if end
  of string reached, continue from beginning of
  string
• sequence of cities in 2nd parent (from 2nd cut
  point) is
• 9-3-4-5-2-1-8-7-6 (remove 4,5,6,7 which are in
  1st offspring)
• 9-3-2-1-8
• place into first offspring o1 (2 1 8 4 5 6
  7 9 3)
• similarly the 2nd offspring o2 (3 4 5 1 8 7 6
  9 2)

Performance Analysis of Meta-Heuristic Approaches
for Traveling Salesperson Problem
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Insertion Mutation

• randomly selects a city
• removes it and inserts it at random position
• Example (1 2 3 4 5 6 7 8 9)
• 4 is selected randomly and placed after 7
• new tour becomes (1 2 3 5 6 7 4 8 9)

Performance Analysis of Meta-Heuristic Approaches
for Traveling Salesperson Problem
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Swap Mutation

• randomly selects two cities in tour
• exchange them
• Example (1 2 3 4 5 6 7 8 9)
• 3rd and 5th selected randomly
• new tour becomes (1 2 5 4 3 6 7 8 9)
• also known as exchange mutation, point mutation,
  reciprocal exchange mutation, order based
  mutation

Performance Analysis of Meta-Heuristic Approaches
for Traveling Salesperson Problem
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Replacement Strategy

• There are variety of strategies for replacing the
  old population by the new offspring population to
  form the next generation
• (Trans-)Generational GA
• N parents produce N (or ?N) offspring (largest
  generation gap,i.e. N).
• Steady-State GA
• Two offspring replace two individuals from the
  old generation

Performance Analysis of Meta-Heuristic Approaches
for Traveling Salesperson Problem
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Termination Criteria

• If the result is known, whenever the population
  converges, i.e. the fitness of the best
  individual in the population is same as the
  expected result, then terminate
• Terminate whenever a predetermined value for the
  number of generations is exceeded
• Terminate whenever all the individuals become
  alike
• Terminate whenever the best individual in the
  population does not change for a long time

Performance Analysis of Meta-Heuristic Approaches
for Traveling Salesperson Problem
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Memetic Algorithms

• Developed by Michael G. Norman and Pablo Moscato
  as a hybrid algorithm
• They combine local search heuristics with
  crossover operators
• In this project Hill Climbing step is applied on
  the children, right after the mutation is
  completed

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for Traveling Salesperson Problem
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Test Datas
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for Traveling Salesperson Problem
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• TEST RESULTS

Performance Analysis of Meta-Heuristic Approaches
for Traveling Salesperson Problem
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TSP of TURKEY
HAKKARI, SIRNAK, SIIRT, BITLIS, MUS, BINGÖL,
ERZINCAN, TUNCELI, ELAZIG, DIYARBAKIR, BATMAN,
MARDIN, SANLIURFA, ADIYAMAN, MALATYA,
KAHRAMANMARAS, GAZIANTEP, KILIS, HATAY (Antakya),
OSMANIYE, ADANA, KAYSERI, YOZGAT, NEVSEHIR,
NIGDE, IÇEL(Mersin), KARAMAN, ANTALYA, BURDUR,
AFYON, ISPARTA, KONYA, AKSARAY, KIRSEHIR,
KIRIKKALE, iKARABÜK, BARTIN, ZONGULDAK, BOLU,
DÜZCE, SAKARYA(Adapazari), BILECIK, KÜTAHYA,
USAK, DENIZLI, MUGLA, AYDIN, IZMIR, MANISA,
BALIKESIR, ÇANAKKALE, EDIRNE, KIRKLARELI,
TEKIRDAG, ISTANBUL, BURSA, YALOVA,
KOCAELI(Izmit), ESKISEHIR, ANKARA, ÇANKIRI,
KASTAMONU, ÇORUM, SINOP, AMASYA, SAMSUN, TOKAT,
SIVAS, ORDU, GIRESUN, GÜMÜSHANE, TRABZON,
BAYBURT, ERZURUM, RIZE, ATRVIN, ARDAHAN, KARS,
AGRI, IGDIR, VAN
Performance Analysis of Meta-Heuristic Approaches
for Traveling Salesperson Problem
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Conclusion

• Considering path representation, several
  different techniques and well known operators are
  tested utilizing 2OPT, Simulated Annealing,
  Genetic Algorithms, Memetic Algorithms and
  Memetic Annealing to solve some small instances
  of Traveling Salesman Problem in 2D, combining
  with hill climbing. Empirical results yield the
  success of the following operators from the best
  to the worst Memetic Annealing performance.
  Performance difference between ISM and classical
  swap EM operator can be observed easily,
  especially on fractal data. Hill climbing
  improved the simulated annealings success rate
  without considering any operator. Since the
  simulated annealing is interested in the
  individual not the population it performs faster
  than genetic algorithms. For the genetic
  algorithmic part OX1, 2PTX and PMX as crossover,
  EM and ISM as mutation and hill climbing method
  performs best. Note that 2PTX uses a patch-up
  algorithm, producing a modified crossover
  combining OX1 and scramble mutation that has not
  been tested before. Hence, the best combination
  of operators is OX1 and EM for both TGGA and
  SSGA. Furthermore, results show that hill
  climbing improves the solution in any GA type.
  2OPT method is the fastest but the poorest one
  when considered with average success rate for an
  overall 100 runs. Since in some parts of the path
  changing of two ways might not be enough so that
  it gets stuck local optima and cannot perform
  better than that. But on the other hand, it would
  be very suitable if anyone looking for a local
  optima instead of a global. It is the cheapest
  and the fastest method for achieving that.

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for Traveling Salesperson Problem
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• DEMOSTRATION

Performance Analysis of Meta-Heuristic Approaches
for Traveling Salesperson Problem